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226. Simple Harmonic Motion.  When a particle moves in a straight line under the action of a force which is directed towards a fixed point and the magnitude of which varies directly as the distance of the particle from the fixed point, the motion is said to be simple harmonic.
Let 0, Fig. 130, be the fixed point, m, the mass of the particle, and x its distance from 0; then the foregoing definition gives
O             m*.
FIG. 130.
where k is the constant of proportionality. The negative sign in the right-hand member of the equation (I7) accounts for the fact that F is directed towards the fixed point, while x is measured in the opposite direction. Substituting this; expression for F in the force equation we get
m|=-fe,                   (I)
QM             2~                                      /T//\
or                                  di=~"X>                          (I)
where co2 =  .    Substituting v~ for -j- in equation (I") and m                         ax       at
integrating we have
v* = c2 - V.
Let v = VQ when x = 0, then c = v0.   Therefore